Geometry: Theorems

Table of Contents

Terms

Throughout our study of geometry in Geometry1 and the first
three SparkNotes of the Geometry2 series, we've essentially built up a library
of knowledge about the kinds of figures that compose geometry. These include
the building blocks, various
constructions, and shapes like polygons
and circles. In
addition, we've looked at three-dimensional
surfaces and the different ways to measure
polygons. The last topic dealt with the concepts of
congruence and
similarity and the consequences inherent when
triangles or certain parts of triangles are congruent or similar. In
congruence, we looked at the techniques for proving that the triangle as a whole
was either congruent or similar. A major part of doing so, we learned, involves
analyzing a figure and determining which parts, if any, are either congruent,
proportional, or neither. Only then, when enough is known about certain parts,
can one of the techniques for proving congruence be used. We already know a few
of these methods. For example, we know that a perpendicular bisector is
perpendicular to the segment it bisects, and intersects that segment at its
midpoint. This fact, along with the others we have learned, are related in that
they are all true by definition.

In the following SparkNote, we'll learn some of the more complex relationships
between parts of figures. These facts are known as theorems. The basic
theorems that we'll learn have been proven in the past. The proofs for all of
them would be far beyond the scope of this text, so we'll just accept them as
true without showing their proof. Eventually we'll develop a bank of knowledge,
or a familiarity with these theorems, which will allow us to prove things on our
own. After the following lessons, we'll recap
everything we know about certain shapes, every relationship between parts, every
fact that is true by definition--everything in our knowledge bank of figure
analysis. With the tools you already have learned, along with those you're
about to learn, you'll be able to conclude a surprisingly great amount from a
figure about which you were told very little. This process of proving
statements geometrically is one of the most important goals of geometry. For
now, we'll only prove things informally. In the SparkNotes making up Geometry3
we'll learn how to do formal proofs.